Finding orientation of a point with respect to another point in xy plane

Question

I have a problem (and a solution too). What I want to know is how to arrive at "that" solution. "That" solution is perfect and working fine for all possible combinations.

Assume that you have 3 points in the XY coordinate system: P1(x1, y1), P2(x2, y2) and P3(x3, y3).

Now, join the point P2 with P3. It will result in "ray" originating from point P2 and passing through point P3. I used word ray because I want it in only one direction, i.e. from P2 to P3.

Now, with respect to point P1, is the ray P2 --> P3 in a clockwise or counterclockwise direction?

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Solution is:

use following formulae to find value of z1:

z1 = (x3 - x1)(y2 - y1) - (y3 - y1)(x2 - x1)

If z1 is positive, P2 --> P3 is clockwise. If z1 is negative, P2 --> P3 is counterclockwise. And if it is 0 the points are on the same imaginary line extending from P1.

Can someone please help me how to arrive at this solution?

Answer 1

The formula you've written is very similar to the formula of cross product of two vectors. As the direction of the cross product is dependent on the CW/CCW configuration, you can easily use it for your problem.

You can construct two rays P1->P2 and P1->P3. Then you can take their cross product. If the component of the product along Z-axis is positive, then P2 and P3 is in counterclockwise order, and vice versa.

If you try to do that, the result, which would be the coefficient of k (unit vector along Z-axis) in the cross product, will be precisely the same as the answer you mentioned.

Answer 2

I am not sure this works. I tried it, but it keeps giving me the same answer for both. Try these coordinates:

Start point 0, 90, 0
Center point 0, 0, 0
End point 90, 0, 0
.....
directional vector 1, 0, 0
inverse directional vector -1, 0, 0